When in weight paint, armature is outlined in red rather than selecting a bone?
First check that your armature is in Pose Mode. Switch it to Pose Mode (Ctrl+Tab) and then select your mesh and paint.

If it doesn's help, you probably didn't assign Armature Modifier to your mesh. In this case select your mesh, then holding Shift select your armature and press Ctrl+P and then choose Armature deform.